This paper introduces a novel two-stage machine learning-based surrogate
modeling framework to address inverse problems in scientific and engineering
fields. In the first stage of the proposed framework, a machine learning model
termed the "learner" identifies a limited set of candidates within the input
design space whose predicted outputs closely align with desired outcomes.
Subsequently, in the second stage, a separate surrogate model, functioning as
an "evaluator," is employed to assess the reduced candidate space generated in
the first stage. This evaluation process eliminates inaccurate and uncertain
solutions, guided by a user-defined coverage level. The framework's distinctive
contribution is the integration of conformal inference, providing a versatile
and efficient approach that can be widely applicable. To demonstrate the
effectiveness of the proposed framework compared to conventional single-stage
inverse problems, we conduct several benchmark tests and investigate an
engineering application focused on the micromechanical modeling of
fiber-reinforced composites. The results affirm the superiority of our proposed
framework, as it consistently produces more reliable solutions. Therefore, the
introduced framework offers a unique perspective on fostering interactions
between machine learning-based surrogate models in real-world applications.

提出了一种新颖的两阶段机器学习的代理建模框架，用于解决科学和工程领域中的反问题。该框架的独特贡献在于集成了符合推理，提供了一种多功能且高效的方法，可以广泛应用。通过进行基准测试和研究集中在纤维增强复合材料的微机械建模的工程应用，结果证实了我们提出的框架的优越性，因为它始终产生更可靠的解决方案。因此，引入的框架为促进机器学习的代理模型在实际应用中的相互作用提供了独特的视角。

基于数据驱动的设计优化的二阶段代理建模及应用于复合微结构生成

Two-Stage Surrogate Modeling for Data-Driven Design Optimization with  Application to Composite Microstructure Generation

The use of machine learning techniques to homogenize the effective behavior
of arbitrary microstructures has been shown to be not only efficient but also
accurate. In a recent work, we demonstrated how to combine state-of-the-art
micromechanical modeling and advanced machine learning techniques to homogenize
complex microstructures exhibiting non-linear and history dependent behaviors.
The resulting homogenized model, termed smart constitutive law (SCL), enables
the adoption of microstructurally informed constitutive laws into finite
element solvers at a fraction of the computational cost required by traditional
concurrent multiscale approaches. In this work, the capabilities of SCLs are
expanded via the introduction of a novel methodology that enforces material
symmetries at the neuron level, applicable across various neural network
architectures. This approach utilizes tensor-based features in neural networks,
facilitating the concise and accurate representation of symmetry-preserving
operations, and is general enough to be extend to problems beyond constitutive
modeling. Details on the construction of these tensor-based neural networks and
their application in learning constitutive laws are presented for both elastic
and inelastic materials. The superiority of this approach over traditional
neural networks is demonstrated in scenarios with limited data and strong
symmetries, through comprehensive testing on various materials, including
isotropic neo-Hookean materials and tensegrity lattice metamaterials. This work
is concluded by a discussion on the potential of this methodology to discover
symmetry bases in materials and by an outline of future research directions.

采用机器学习技术同化任意微观结构的有效行为已被证明不仅高效而且准确。本文结合先进的微观力学模型和机器学习技术展示了如何同化表现非线性和历史相关行为的复杂微观结构，通过所得的智能本构法（SCL）将微观信息融入有限元求解器，所需计算成本仅为传统并发多尺度方法的一小部分。本文通过引入在神经元级别强制施加材料对称性的新方法，扩展了 SCL 的能力，适用于各种神经网络架构。该方法利用基于张量的特征在神经网络中有效、准确地表示保持对称操作，且泛化性足够强以应用于超越本构模型的问题。详细介绍了构建这些基于张量的神经网络以及其在学习本构法中的应用，包括对弹性材料和非弹性材料的测试。通过对各种材料（包括各向同性的新胡克材料和张力结构格子异质材料）进行全面测试，验证了该方法在数据有限且存在强对称性的情况下优于传统神经网络的优越性。最后讨论了该方法在材料对称基础发现和未来研究方向的潜力。